The objective of this application is to continue developing improved methods of analysis of disease incidence data in family studies. Issues to be addressed include censored age-at-onset outcomes, major genes, polygenes, and shared unmeasured environmental factors, linkage to multiple polymorphic markers, and associations with candidate genes, genetic heterogeneity, measured environmental exposures with gene-environment interactions, misclassified or missing data, and correction for ascertainment. The basic model being considered is an extension of the proportional hazards model which includes latent variables for each individual representing their unobserved genotypes. A Monte Carlo technique, known as Gibbs sampling, and generalized estimating equations (GEE) methods are used to overcome the considerable computational burden that would otherwise be required in a full likelihood treatment. The specific aims of this application have been extended beyond those of the original application. The investigators have omitted the previous aim for development of computer software and have added two aims. The specific aims include 1) further development of the methods, with priority given to extensions to multi-locus mapping, genetic heterogeneity, ascertainment correction, and Markov chain Monte Carlo theory; 2) further development of the generalized estimating equations approach to segregation analysis extending it to analysis of binary and survival data, and to ascertainment through probands; 3) exploration of statistical design issues in family studies aimed to segregation, linkage, candidate gene, and gene environment interaction analyses; 4) simulation studies of the performance of the methods compared with currently available methods in genetics and epidemiology, including assessment of the power to distinguish alternative genetic models and patterns of gene environment interactions, the robustness of the methods to misclassification of the model or the ascertainment protocol, and the relative efficiency of alternative designs; and 5) applications to various family studies the investigators are conducting, including breast and ovarian cancer, colorectal cancer, and diabetes.